PCAn {PTAk}

Principal Component Analysis on n modes

Description

Performs the Tuckern model using a space version of RPVSCC (SINGVA).

Usage

 PCAn(X,dim=c(2,2,2,3),test=1E-12,Maxiter=400,
              smoothing=FALSE,smoo=list(NA),
                verbose=getOption("verbose"),file=NULL,
                  modesnam=NULL,addedcomment="")

Arguments

X

a tensor (as an array) of order k, if non-identity metrics are used X is a list with data as the array and met a list of metrics

dim

a vector of numbers specifying the dimensions in each space

test

control of convergence

Maxiter

maximum number of iterations allowed for convergence

smoothing

see SVDgen

smoo

see PTA3

verbose

control printing

file

output printed at the prompt if NULL, or printed in the given ‘file’

modesnam

character vector of the names of the modes, if NULL "mo 1" ..."mo k"

addedcomment

character string printed after the title of the analysis

Details

Looking for the best rank-one tensor approximation (LS) the three methods described in the package are equivalent. If the number of tensors looked for is greater then one the methods differs: PTA-kmodes will "look" for "best" approximation according to the orthogonal rank (i.e. the rank-one tensors are orthogonal), PCA-kmodes will look for best approximation according to the space ranks (i.e. the rank of every bilinear form, that is the number of components in each space), PARAFAC/CANDECOMP will look for best approximation according to the rank (i.e. the rank-one tensors are not necessarily orthogonal). For the sake of comparisons the PARAFAC/CANDECOMP method and the PCA-nmodes are also in the package but complete functionnality of the use these methods and more complete packages may be fetched at the www site quoted below.
Recent work from Tamara G Kolda showed on an example that orthogonal rank decompositions are not necesseraly nested. This makes PTA-kmodes a model with nested decompositions not giving the exact orthogonal rank. So PTA-kmodes will look for best approximation according to orthogonal tensors in a nested approximmation process.

Value

a PCAn (inherits PTAk) object

Note

The use of metrics (diagonal or not) and smoothing extend flexibility of analysis.

Author(s)

Didier G. Leibovici

References

Caroll J.D and Chang J.J (1970) Analysis of individual differences in multidimensional scaling via n-way generalization of "Eckart-Young" decomposition. Psychometrika 35,283-319.

Harshman R.A (1970) Foundations of the PARAFAC procedure: models and conditions for "an explanatory" multi-mode factor analysis. UCLA Working Papers in Phonetics, 16,1-84.

Kroonenberg P (1983) Three-mode Principal Component Analysis: Theory and Applications. DSWO press. Leiden. (There was a maintained (by Pieter) library of contributions to multiway analysis ...))

Leibovici D and Sabatier R (1998) A Singular Value Decomposition of a k-ways array for a Principal Component Analysis of multi-way data, the PTA-k. Linear Algebra and its Applications, 269:307-329.

Kolda T.G (2003) A Counterexample to the Possibility of an Extension of the Eckart-Young Low-Rank Approximation Theorem for the Orthogonal Rank Tensor Decomposition. SIAM J. Matrix Analysis, 24(2):763-767, Jan. 2003.